Non-equilibrium dynamics of dipolar polarons

Dear Editors of SciPost Physics,

Thank you for handling our submission and for the invitation to revise and resubmit the manuscript.

Both Referees found that our work can be suitable for publication provided we address their comments. We revise our manuscript following the Referee's reports and provide a point-by-point reply.
For convenience, we highlight the main changes in the manuscript. Minor changes (correction of typos and style) are not highlighted.

We hope that our manuscript is now ready for publication in SciPost Physics.

Sincerely,

The Authors

\section*{\bf Reply to Reviewer 1}

{\bf Reviewer }: This manuscript presents a perturbation study of the polaron physics of a weakly interacting impurity immersed in a dipolar Bose-Einstein condensate. The authors employ the time-dependent variational theory on the polaron Hamiltonian in the moving frame of the polaron, obtained through the Lee-Low-Pines transformation. The paper explores various aspects of the polaron, such as steady-state energy, residue, effective mass, and relaxation dynamics. Additionally, the authors investigate the trapped system using a local-density approximation. Overall, the topic is interesting, and the manuscript is well-written and provides clear motivations, explanations, and detailed analysis. However, there are some concerns regarding the formalism that should be addressed before making a final recommendation.

{\bf Our reply:} We thank the Referee for taking the time to review our work, for a positive evaluation of our results and for suggestions that helped us to improve the manuscript considerably. We think we have addressed all of the concerns listed in the report and hope that the improved version of the manuscript is suitable for publication.

{\bf Reviewer }: Derivation of Eqs. (2):
It would be beneficial to include a more detailed derivation of Eqs. (2) from the time-dependent variational ansatz that minimizes the action functional. Specifically, it would be helpful to explain why the second term on the right-hand side of the second equation in Eqs. (2) is not expressed as $(P - P_B)^2/2m$, given the naive expectation that it should resemble the Hamiltonian in Eq. (S4).

{\bf Our reply:} We thank the Referee for this comment. In the revised manuscript, we have added a detailed derivation of Eq. (2), see new App. C.

{ \bf Reviewer:}
Neglect of $P_B$ in the perturbation analysis:
The authors state that in the Lee-Low-Pines transformation, $P_B$ (which is related to the recoil of the impurity by the phonons) is neglected in their perturbation analysis because it only enters in the next-to-leading order. However, neglecting $P_B$ implies an effective Hamiltonian equivalent to that of an infinitely heavy impurity. This contradicts the assumption of equal masses for the impurity and background bosons, where recoil should play a significant role. It is necessary to provide a more detailed discussion justifying the neglect of $P_B$ and its underlying physical picture.

{\bf Our reply:} We respectfully disagree with the Referee that the limit $\mathbf{P}_{\mathrm{B}}=0$ corresponds to an infinitely heavy impurity. The mass of the impurity enters in our equations even after we set $\mathbf{P}_{\mathrm{B}}=0$, see Eq. (2), which contains the parameter $m$.

We agree with the Referee that the boson-impurity scattering plays an important role. In particular, it leads to renormalization of the impurity quasiparticle properties, such as energy and mass, which we calculate and benchmark against available results in the literature. It also affects dynamic properties as we discuss in the text.

To further disentangle the limit $\mathbf{P}_{\mathrm{B}}=0$ from the limit of an infinitely heavy impurity, please note that for the ground state with $\mathbf{P}=0$ it is natural to expect (as there is no vector to set a preferred direction) that $\mathbf{P}_{\mathrm{B}}=0$ for any mass of the impurity. However, even in this case, the mass of the impurity enters our calculations as it determines the energy-momentum conservation laws for boson-impurity scattering. In the revised manuscript, we added a clarifying discussion; see the last paragraph in new App.~C and new Eq. (8) in the main text.

{\bf Reviewer :}
Self-trapping effect and earlier works:
Considering that the authors focus on weakly interacting impurities, it would be valuable to mention early works on ultracold polarons that consider the self-trapping effect. The self-trapping effect leads to the spontaneous breaking of spatial translational symmetry, and the momentum P is no longer a good quantum number. Although the mean-field description fails for strongly interacting impurities, it is likely that the self-trapping effect remains relevant for weakly interacting impurities, at least for static state analysis. (I would expect that for dynamical analysis, this is less relevant, as the weakly-interacting BEC is a slowly-responded fluid.) This effect becomes particularly interesting in systems with long-range interactions. It would be appropriate to reference earlier works such as [Phys. Rev. Lett. 96, 210401 (2006), Phys. Rev. A 73, 043608 (2006)] and relevant literature in this context.

{\bf Our reply:} We thank the Referee for this comment. Indeed, we have not discussed self-localization of the impurity in our work. In the revised manuscript, we argue that self-localization is not relevant for the dilute gas that we consider, see the discussion at the end of App. B. The main reason for that is that the gas parameter, $n a_{11}^3$, is very small for our parameters.

{\bf Reviewer:}
Some Minor Suggestions:

1. Discuss the underlying physics of relaxation:
Since there are no decay channels mentioned, it would be helpful for the authors to comment on the underlying physics of the relaxation process. It could be speculated that the relaxation is caused by some dephasing process. Providing insights into this aspect would enhance the understanding of the dynamics studied.

2. Perturbation argument in droplet or supersolid backgrounds:
While the authors claim that the results can be extrapolated easily to droplet or supersolid backgrounds, it is important to note that the density in such regimes might be significantly higher than in a Bose-Einstein condensate. Consequently, the validity of the perturbation argument may no longer hold. It would be fair to mention this limitation to the readers in the Outlook section.

{\bf Our reply:} 1. We thank the Referee for this suggestion. As the Referee correctly pointed out, the underlying mechanism of relaxation is a dephasing process. The systems relaxes even in the absence of boson-boson interactions because the rapid change of the impurity-boson interaction populates a continuum of energy states, which then dephase due to the energy mismatch, similar to Refs. [21, 28] of our manuscript.

Importantly, in real experiments, other dephasing relaxation processes are possible. They can
originate due to trap dephasing, decoherence by magnetic-field fluctuations, three-body losses and thermal fluctuations. To compare to our results, these decoherence effects should be under control in cold-atom experiments (cf.~Ref.~[10] of the revised manuscript).\\

We have added a clarifying discussion, see footnote 2 of the revised manuscript.\\


2. We agree with the Referee that the density of quantum droplets is higher than the one we consider in the paper.
Quantum fluctuations or in our language, the beyond-mean-field (Lee-Huang-Yang) contribution is the main mechanism for the stabilization of quantum droplets. This term may become important for densities one order of magnitude larger than those typical for BECs (namely $\sim10^{21}/\mathrm{m}^{3}$). {\it However}, it should be noted that perturbative approaches such as Gross–Pitaevskii-like equations have been widely used and their accuracy have been confirmed by comparing to experimental data in the droplet and supersolid regimes. See, for example, the reference R.N. Bisset, et.al., Phys. Rev. Lett. 126, 025301 (2021), where the polaron energy was obtained from second-order perturbation theory and benchmarked against the chemical potential of the impurity immersed in a dipolar quantum droplet.

{\bf Reviewer:} Overall, the manuscript is well-written and addresses an interesting topic. By addressing the concerns raised and incorporating the suggestions provided, the authors can further improve the paper and make it suitable for publication.

{\bf Our reply:} We thank these encouraging remarks on our work. We hope that the revised version of our manuscript is ready for publication in SciPost Physics.

\section*{\bf Reply to Reviewer 2}

{\bf Reviewer }:
In this work, the authors study the non-equilibrium dynamics of dipolar Bose polarons after an interaction quench of the impurity-boson coupling. Certain dynamical features of this system are calculated using a time-dependent variational approach and exploiting the Lee-Low-Pines transformation which allows also for analytical estimates and for distinguishing the relevant timescales. They extract several experimentally relevant observables such as the Ramsey contrast and quasiparticle effective mass. The main result is the manifestation of the polaron anisotropic relaxation dynamics due to the dipolar interactions. Trap effects are also explored within the local density approximation.

{\bf Our reply:} We thank the Referee for taking the time to review our work.

{\bf Reviewer }:
I find these results interesting towards the understanding of polarons in anisotropic media. Moreover, the submission is timely and the setup will be accessible in near future cold atom experiments. The manuscript is also well written. However, I have some comments and questions regarding the presentation and the interpretation of the presented results. If the authors are able to address these concerns which I provide below, the work will be further advanced and be suitable for publication in SciPost Physics. Suggestions for improvement follow

{\bf Our reply:} We thank the Referee for this positive evaluation of our work, and for comments that helped us to improve the manuscript. We hope that the Referee will find that the revised manuscript addresses all of the comments listed in the report, making our work suitable for publication in SciPost Physics.

{\bf Reviewer}:
In the introduction, the statement “collective excitations build the subsequent time evolution” is not clear. What type of collective excitations are meant and in which context? I guess the few-body correlations should also play a role at long evolution times invalidating the mean-field approximation? Please elaborate at least briefly on this point.

{\bf Our reply:} We thank the Referee for identifying this weak statement. To avoid any confusion, we have re-written this sentence. In particular, we have decided to refrain from discussing collective excitations that early in the paper. The goal of our introduction is to provide a general motivation for studying non-equilibrium dynamics in general setting and various collective excitations. \\

To explain the importance of few- and many-body correlations, we now add a discussion to the beginning of the third paragraph of the Introduction: \textit{Complexity of quench dynamics of impurities stems in particular from the interplay between
few- and many-body correlations and the corresponding timescales. Initial dynamics involves
a local excitation of the system, which implies participation of phonons with high momenta that can resolve few-body physics. After some time, the dynamics is mainly
driven by the propagation of low-energy many-body excitations}. Of course, the Referee is right that the system always has few- and many-body correlations. By discussing few- and many-body correlations, we want to stress that the initial time dynamics can be modeled using mainly few-body physics, while many-body correlations become more important at the later stages of time evolution.

{\bf Reviewer}:
In the introduction the term short-time polaron physics is stated. This becomes particularly confusing since a sentence later the term initial short time dynamics is used. Please provide an order of magnitude of what is actually meant.

{\bf Our reply:}
We thank the Referee for identifying this possibly confusing issue. In fact, short-time dynamics and initial short-time are used in the same context. In order to avoid any confusion, we have rewritten this part of the introduction. In particular, we have decided to avoid introducing timescales in the introduction. They are now introduced later in the main text.\\

{\bf Reviewer }:
In the same paragraph, it would be beneficial for the reader to already explain in a sentence the context of the Ramsey contrast. Otherwise, the comment provided at the end of the Introduction should be moved to this point and explain which interactions are meant in the overlap.

{\bf Our reply:} Following the recommendation of the Referee, we have added a discussion about the Ramsey-type scheme to the introduction: {\it In particular, a combination of a RF pulse that coherently populates the impurity state and Ramsey
interferometry, which tracks down the quantum evolution of the impurity coupled coherently to
the BEC, allows one to measure a time-dependent Green’s function, which contains information about
non-equilibrium dynamics of the impurity [9, 10]. Its imaginary part gives the contrast, namely the
overlap between the initial state and the polaron state, which is one of the key observables studied
in this work.}

{\bf Reviewer }: Since the authors provide a list of possible techniques that have been used to describe polaron physics it would be fair to also comment on other variational methods that have been extensively employed.

{\bf Our reply:} In the revised manuscript, we have added a number of references in the introduction and throughout the text. In particular, the revised version of the introduction now mentions the Langevin equation, the many-body generalization of Weisskopf-Wigner theory, and a few papers for an impurity in a Fermi gas. In general, following the recommendation of the Referees, we have extended our bibliography significantly by adding a number of relevant works.

{\bf Reviewer}:
I would also encourage the authors to describe their main specific results in a more clear manner in the Introduction.

{\bf Our reply:} Following this recommendation of the Referee, we have added a corresponding discussion as the last two paragraphs of the introduction.

{\bf Reviewer}:
I wonder whether in the single impurity limit that is considered in Section 2, dipole-dipole interactions between the impurity and the bath are indeed experimentally justified. Please comment.

{\bf Our reply:} In experiments, a small number of impurities are created on top of the condensate. In general, the impurity concentration is lower than ten percent. The direct and induced interactions between impurities are (almost) negligible compared to boson-impurity interactions. Thus, the ultra-dilute gas of impurities is well described within the framework of a single impurity. For instance, in dipolar mixtures such as $\mathrm{Dy-Dy}$, $\mathrm{Er-Dy}$ or $\mathrm{Er-Er}$, the impurity-boson dipolar interaction is proportional to the dipolar length, $d_2$. It is always present, and it prominently enters our analytical expressions.

{\bf Reviewer }:
The fact that the relaxation timescales are less for a head-to-to-tail arrangement should be explained. Is it related to the attractive nature of the interaction potential and how? Please comment. The term “relaxation time” should also be explicitly explained since it has different interpretations in the literature.

{\bf Our reply:}
The referee is right that the relaxation timescales are shorter for a head-to-to-tail arrangement. In order to explain the previous statement, we added a few sentences in the main text and in the appendix.\\

In the main text, we added:\\

\textit{In the absence of dipolar interactions, polaron formation time, which marks the onset of the final stage of the impurity dynamics, is given by $\xi/c_0$~[25], where $\xi=\hbar/(\sqrt{2}c_0)$ is the coherence length of a non-dipolar condensate. If we naively define a direction-dependent timescle as $\hbar^2/(\sqrt{2}c(\theta)^2)$ (see App.~D for a derivation of this timescale), then we shall expect slower dynamics in the side-to-side ($\theta=\pi/2$) direction, which features the softest excitations in the system~[50], than in the head-to-tail ($\theta=0$) one.
This expectation is directly confirmed by our calculations, see Fig.~1 (b).}\\

In App. D, after Eq.~[S32], we added:\\

\textit{For a non-dipolar condensate, $t_n$ is the time required for a phonon to travel through the region of the condensate distorted by the impurity. Indeed, $t_n\simeq \xi/c$, where the healing length $\xi$ defines the size of the distortion due the presence of the impurity. For a dipolar condensate, the anisotropy of time evolution is encoded in the prefactor $\bar{\Omega}_{x}$, which effectively leads to a direction-dependent speed of sound.}

{\bf Reviewer }:
It would be interesting to comment whether quantum fluctuations (in terms of second order perturbation) play any role in the static (e.g. ground state) polaron energy for the considered interactions. In other words, are there any deviations from the mean field contribution?

{\bf Our reply:} Following the recommendation of the Referee, we have added an estimate of the contribution to the ground-state energy from second-order perturbation theory. This estimate is about 10\%.

{\bf Reviewer }:
In the contrast defined in Equation (3), I guess the initial wave function refers to the situation where all interactions are switched-off. Is this correct? Please comment explicitly.

{\bf Our reply:} Indeed, the initial state corresponds to a system where the boson-impurity interaction is turned off, and the total momentum is vanishing.
We have added a corresponding explanation to the revised manuscript, see Sec. 4.2. It is worth noting also that the most relevant case for current experiments is to consider a non-dipolar impurity immersed in a dipolar bath. In this case, only the s-wave impurity-boson scattering length is switched on.

{\bf Reviewer}:
What is the physical context and the formula of the timescale $t_n$ appearing in Equation (4)? How is it compared to $t_\Omega$?

{\bf Our reply:} The parameter $t_n=\frac{m}{8\pi n\hbar a_{11}}$ defines the timescale for the Bose gas at the mean-field level. Following the recommendation of the Referee, we add this explicitly to the manuscript, see the discussion around Eq.~4. For the considered parameters, $t_n\sim 0.1$ms, whereas $t_\Omega\sim 0.2$s, see the caption to Fig.~2. It is worthnoting that the interplay between the parameters $t_\Omega$ and $t_n$ is seen throughout our work, see for example Eq. (5) and Eq. (14) for an experimentally relevant trapped case.

{\bf Reviewer}:
In dipolar systems it is known that loss channels such as three-body recombination play an important role. Can the authors comment on the effect of such processes in their system.

{\bf Our reply:}
We agree with the Referee that
three-body recombination processes play an important role in bosonic systems. However, note that the system under consideration is very weakly interacting, meaning that it survives in a trap for long times that can be of the order of 100ms and even more, (see, e.g., Phys. Rev. Lett. 121(21), 213601 (2018)). As typical time scales for polaron formation are less than 1ms, we do not expect three-body processes to play an important role in our work.

To further address the comment of the Referee, we note that the processes involving recombination of one impurity and two bosons from the condensate yield a distribution of atoms lost from the trap, which is proportional to the number of atoms transferred into the impurity state. These effects can be indeed very important for large interspecies interaction. For example, they are probably relevant for $^{166}$Er-$^{162}$Dy mixtures (see, e.g., Phys. Rev. Lett. 121(21), 213601 (2018)). As in the manuscript, we focus on the scattering lengths $\simeq d$, we can neglect these dynamics here.

{\bf Reviewer}:
The decay of the contrast shown in Figure is relatively weak. Would it be possible to be observed in the experiment? Can the authors provide their opinion on this issue?

{\bf Our reply:} We agree with the Referee that the change of the contrast is weak in the uniform case. However, in real experiments, trap dephasing is essential for time dynamics, and therefore, the situation presented in Fig.~3 of our manuscript cannot be observed in current experiments. Note that in the trapped case, the contrast ranges from 0 to 1 and it is accessible via interferometry from very short time scales up to the steady state using already available experimental techniques.

{\bf Reviewer}:
Does the system exhibits any universal characteristics for long evolution times where e.g. the contract is saturated?

{\bf Our reply:} In Eq.~(9), we present an expression for the contrast for longer times. The expression depends on the compressibility of the bath encoded in the gas parameter $na_{11}^{3}$, the dipolar coupling strengths and the s-wave impurity boson scattering length. Therefore, it seems that there is no universality, as one cannot toss out any of the relevant length scales. Similarly to the non-dipolar case, universality is only unveiled for short-time dynamics, see Ref.~[10] of our work

{\bf Reviewer }:
I expect that depending on the $\epsilon_{dd}$ parameter the phase of the bath is modified. How is this fact be reflected in the contrast?

{\bf Our reply:} Indeed, by changing the parameter $\epsilon_{dd}$ one can modify the phase of the host bath. However, as we focus in our work on the regime with $\epsilon_{dd}<1$, the system is still a BEC with a well-defined global phase. This phase is insensitive to the presence of the impurity. We leave a more detailed investigation on the role of the phase of the bath to future studies.

{\bf Reviewer }:
I would expect that the diffusion process is accompanied by energy redistribution between the impurity and the medium. Can the authors comment on this issue?

{\bf Our reply:} It is true that quench dynamics is accompanied by energy redistribution between the impurity and the medium. To clarify the dynamics of the energy we have modified the text, see the last part of new App. C and new Eq. (8).

{\bf Reviewer }:
Does the fact that the impurity slowdown occurs faster for $\epsilon_{dd} \to1$ is related to the superfluid character of the medium or not? Please clarify.

{\bf Our reply:} The notion of `critical slowdown' requires a superfluid character of the medium -- it defines the behavior of the system when the momentum of the impurity is close to the critical momentum, see Ref. [25] of our manuscript.
When $\epsilon_{dd}\rightarrow1$, the speed of sound in one of the directions vanishes, which means that the slowdown occurs for smaller values of the momentum of the impurity.

We have clarified this discussion in the revised manuscript.

{\bf Reviewer}:
I am not sure that the trap frequencies provided before Equation (11) have been used in dipolar condensates. If they have been used please provide relevant references.

{\bf Our reply:} Although, we have chosen these values of frequencies rather arbitrarily, they are not far from what have been seen with cold dipolar atoms (see, e.g., Phys. Rev. Lett. 121(21),
213601 (2018)). In general, we do not foresee any difficulty in achieving experimentally the proposed numbers due to a high tunability of harmonic trapping in cold-atom laboratories.

{\bf Reviewer }:
In the outlook section, I would suggest adding at least a brief description of the new results presented in the main text. This will certainly facilitate the understanding of the reader.

{\bf Our reply:} Following this recommendation of the Referee, we have briefly described the main results of the paper in the last section, see the beginning of Sec. 4.6 of the revised manuscript.

We thank the Referee for the comments and hope that the revised version of our manuscript is ready for publication in SciPost Physics.